Note that in the case of the Schrödinger operator \(-\Delta +V(x)\) , the Jost functions are known to be very useful tools for studying asymptotic behaviors of the resolvent (see [1]}) and have been widely used in the proof of \(L^p\) -boundedness of wave operators (see [2]}, [3]}, [4]}). However, it is not clear whether the same method can be also applied to the fourth-order case. Indeed, in view of the explicit formula of the free resolvents \((\Delta ^2-\lambda ^4\mp i0)^{-1}\)  (see [freeresolvent]freeresolvent below), we must construct four Jost functions \(f_\pm (\lambda ,x),g_\pm (\lambda ,x)\)  such that
\(f_\pm (\lambda ,x)\sim e^{\pm i\lambda x},\quad g_\pm (\lambda ,x)\sim e^{\mp \lambda x},\quad x\rightarrow \pm \infty ,\) 
